# 4.1 Linear functions  (Page 15/27)

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If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y -intercepts.

If a horizontal line has the equation $\text{\hspace{0.17em}}f\left(x\right)=a\text{\hspace{0.17em}}$ and a vertical line has the equation $\text{\hspace{0.17em}}x=a,\text{\hspace{0.17em}}$ what is the point of intersection? Explain why what you found is the point of intersection.

The point of intersection is This is because for the horizontal line, all of the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates are $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and for the vertical line, all of the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinates are $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ The point of intersection is on both lines and therefore will have these two characteristics.

## Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.

$y=\frac{1}{4}x+6$

$y=3x-5$

Yes

$y=3{x}^{2}-2$

$3x+5y=15$

Yes

$3{x}^{2}+5y=15$

$3x+5{y}^{2}=15$

No

$-2{x}^{2}+3{y}^{2}=6$

$-\frac{x-3}{5}=2y$

Yes

For the following exercises, determine whether each function is increasing or decreasing.

$f\left(x\right)=4x+3$

$g\left(x\right)=5x+6$

Increasing

$a\left(x\right)=5-2x$

$b\left(x\right)=8-3x$

Decreasing

$h\left(x\right)=-2x+4$

$k\left(x\right)=-4x+1$

Decreasing

$j\left(x\right)=\frac{1}{2}x-3$

$p\left(x\right)=\frac{1}{4}x-5$

Increasing

$n\left(x\right)=-\frac{1}{3}x-2$

$m\left(x\right)=-\frac{3}{8}x+3$

Decreasing

For the following exercises, find the slope of the line that passes through the two given points.

$\left(2,4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{10}\right)$

$\left(1,\text{5}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{11}\right)$

2

$\left(–1,\text{4}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{2}\right)$

$\left(8,–2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,6\right)$

–2

$\left(6,11\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(–4,\text{3}\right)$

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

$f\left(-5\right)=-4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=2$

$y=\frac{3}{5}x-1$

$f\left(-1\right)=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=1$

Passes through $\text{\hspace{0.17em}}\left(2,4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,10\right)$

$y=3x-2$

Passes through $\text{\hspace{0.17em}}\left(1,5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,11\right)$

Passes through $\text{\hspace{0.17em}}\left(-1,\text{4}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{2}\right)$

$y=-\frac{1}{3}x+\frac{11}{3}$

Passes through $\text{\hspace{0.17em}}\left(-2,\text{8}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{6}\right)$

x intercept at $\text{\hspace{0.17em}}\left(-2,\text{0}\right)\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}\left(0,-3\right)$

$y=-1.5x-3$

x intercept at $\text{\hspace{0.17em}}\left(-5,\text{0}\right)\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}\left(0,\text{4}\right)$

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.

$\begin{array}{l}4x-7y=10\hfill \\ 7x+4y=1\hfill \end{array}$

perpendicular

$\begin{array}{c}3y+x=12\\ -y=8x+1\end{array}$

$\begin{array}{c}3y+4x=12\\ -6y=8x+1\end{array}$

parallel

$\begin{array}{l}6x-9y=10\hfill \\ 3x+2y=1\hfill \end{array}$

For the following exercises, find the x - and y- intercepts of each equation.

$f\left(x\right)=-x+2$

$\begin{array}{l}f\left(0\right)=-\left(0\right)+2\\ f\left(0\right)=2\\ y-\mathrm{int}:\left(0,2\right)\\ 0=-x+2\\ x-\mathrm{int}:\left(2,0\right)\end{array}$

$g\left(x\right)=2x+4$

$h\left(x\right)=3x-5$

$\begin{array}{l}h\left(0\right)=3\left(0\right)-5\\ h\left(0\right)=-5\\ y-\mathrm{int}:\left(0,-5\right)\\ 0=3x-5\\ x-\mathrm{int}:\left(\frac{5}{3},0\right)\end{array}$

$k\left(x\right)=-5x+1$

$-2x+5y=20$

$\begin{array}{l}-2x+5y=20\\ -2\left(0\right)+5y=20\\ 5y=20\\ y=4\\ y-\mathrm{int}:\left(0,4\right)\\ -2x+5\left(0\right)=20\\ x=-10\\ x-\mathrm{int}:\left(-10,0\right)\end{array}$

$7x+2y=56$

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through $\text{\hspace{0.17em}}\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-24\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-1,19\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,-71\right)$

Line 1: m = –10 Line 2: m = –10 Parallel

Line 1: Passes through $\text{\hspace{0.17em}}\left(-8,-55\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(10,89\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(9,-44\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,-14\right)$

Line 1: Passes through $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,-1\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(6,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,5\right)$

Line 1: m = –2 Line 2: m = 1 Neither

Line 1: Passes through $\text{\hspace{0.17em}}\left(1,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,5\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-1,-3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,1\right)$

Line 1: Passes through $\text{\hspace{0.17em}}\left(2,5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,-1\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-3,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-5\right)$

For the following exercises, write an equation for the line described.

Write an equation for a line parallel to $\text{\hspace{0.17em}}f\left(x\right)=-5x-3\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(2,\text{–}12\right).$

Write an equation for a line parallel to $\text{\hspace{0.17em}}g\left(x\right)=3x-1\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(4,9\right).$

$y=3x-3$

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